Modal solution for the fields due to a magnetic source impressed in a circular cylinder

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Analytic solutions have been developed to solve for the electromagnetic field due to a radiating aperture on the surface of canonical shapes, such as, infinite circular cylinder, spheres, and prolate spheroids. The solutions were expressed in terms of infinite Fourier series of the form SigmaC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> cos(mphi), with C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> as a function of the radius, rho. These solutions were based on harmonic series using integer order Bessel functions. This series is poorly convergent for large arguments of the Bessel functions (when ka Gt 1). Watson [1] developed a method for large values of ka, where he transformed the poor converging harmonic series into a contour integral, deforming its integration path for capturing individual terms called residues of the poles. For the pattern of a slotted cylinder antenna, the residue series proposed by Watson is highly convergent in the direction away from the slot; known as the shadow region. Even though, better results were obtained for the region forward to the slot, or the illuminated region, when using physical and geometrical optics. Watson assumed that there are no surface currents in the shadow region and also approximated the induced current in the illuminated region by the current that would be induced on the local tangent plane. While good results were obtained at the illuminated region, incorrect results were obtained from the shadow and the transition regions. Keller developed the Geometrical Theory of Diffraction (GTD) [2] with the assumption that fields propagate along rays, this theory includes the effects of diffraction which are not considered in the geometrical optics. An extension in the GTD was made by Pathak et al. [3] called Uniform GTD (UTD). This method introduces a new dyadic torsion factor term [3], that improves the solution in the transition region including the boundary with the shadow region, where the GTD failed, and reduces to the geometric optics solution in the lit or illuminated region.